Mathematical Proof of the Mandel--Cryer Effect in Poroelasticity
C.J. van Duijn, Andro Mikelić · Multiscale Modeling and Simulation · 2021
We consider Mandel's problem from poroelasticity, which describes the behavior of a water saturated porous sample being sandwiched between two rigid plates. It was observed, both computationally and experimentally, that the pore pressure in the center of the sample increases for some time and decreases later. This is known as the Mandel--Cryer effect. It is the purpose of this paper to provide a rigorous mathematical setting for Mandel's problem and for the corresponding Mandel--Cryer effect. We first formulate nonstandard linear parabolic problems for the volume strain and the fluid pressure. These problems admit “explicit” solutions in terms of Fourier series. Introducing the abstract variational parabolic formulation with appropriate spaces, the Fourier series are shown to converge strongly. The main result is the mathematical proof of the Mandel--Cryer effect. Here we use the Laplace transform applied to the pressure equation. We write the transformed pressure in such a way, that a Tauberian type of result applies to its time derivative. From this the Mandel--Cryer effect is immediate.